Control of Unsteady Solidification Via Optimized Magnetic Fields

ABSTRACT

This article presents a numerical procedure for automatically controlling desired features of a melt undergoing solidification by applying an external magnetic field whose time-varying intensity and spatial distribution are obtained by the use of a hybrid optimization algorithm. The intensities of the magnets along the boundaries of the container were discretized by using B-splines. The inverse problem is then formulated to find the magnetic boundary conditions (the coefficients of the B-splines) in such a way that the gradients of temperature along the gravity direction are minimized at each instant as the solidification front advances through a moving melt. For this task, a hybrid optimization code was used that automatically switches among the following six optimization modules; the Davidon-Fletcher-Powell (DFP) gradient method, a genetic algorithm (GA), the Nelder-Mead (NM) simplex method, quasi-Newton algorithm of Pshenichny-Danilin (LM), differential evolution (DE), and sequential quadratic programming (SQP). Transient Navier-Stokes and Maxwell's equations were discretized by using a finite volume method in a generalized curvilinear nonorthogonal coordinate system. For the phase change problems, an enthalpy formulation was used. The computer code was validated against analytical and numerical benchmark results with very good agreements in both cases.

Key Words:

• Flow control
• Genetic algorithm
• Hybrid optimization
• Magnetohydrodynamics
• Smart manufacturing
• Solidification

ACKNOWLEDGMENTS

The first author is grateful for the postdoctoral fellowship received from University of Texas at Arlington and from CNPq, a Brazilian council for scientific and technological development. The second author is grateful for the partial support provided for this research on the grant NSF DMS-0073698 administered through the Computational Mathematics program.

The original paper was presented as “Reducing Convection Effects in Solidification by Applying Magnetic Fields Having Optimized Intensity Distribution” (by Colaco, M.J., Dulikravich, G.S. and Martin, T.J.), Keynote Lecture, ASME Paper HT2003-47308, ASME Summer Heat Transfer Conference, Las Vegas, NV, July 21–23, 2003.